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A005489 Number of nonzero coefficients of order n in Baker-Campbell-Hausdorff expansion.
(Formerly M0181)
1
1, 1, 2, 1, 8, 7, 32, 31, 96, 97, 512, 511, 2048, 2047, 7396, 7531, 32768, 32767, 131072, 131071, 508436, 512245, 2097152, 2097151, 8202208, 8207797, 33256980, 33335611, 134217728, 134217727 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..30.

J. A. Oteo, The Baker-Campbell-Hausdorff formula and nested commutator relations, J. Math. Phys., 32 (1991), 419-421.

J. A. Oteo, The Baker-Campbell-Hausdorff formula and nested commutator identities, J. Math. Phys., 32.2 (1991), 419, 421. (Annotated scanned copy)

MATHEMATICA

g[1] = 1;

g[s_] := g[s] = Expand[(2 t - 1) g[s - 1] + t (t - 1) D[g[s - 1], t]];

cx[ss_] := Module[{m, mp, mpp, \[Gamma]},

   m = Length[ss] + 1;

   mp = Floor[m/2];

   mpp = Floor[(m - 1)/2];

   \[Gamma] = CoefficientList[Product[g[s], {s, ss}], t];

   (-1)^mpp mpp! / Product[s!, {s, ss}] Sum[\[Gamma][[k]] (mp + k - 1)!/(m + k - 1)!, {k, Total[ss] - m + 2}]

];

cxs[n_] := Select[Table[{cx[ss], Length@Permutations@ss}, {ss, IntegerPartitions[n - 1]}], First@# != 0 &];

a[n_] := Total[Last /@ cxs[n]];

Table[a[n], {n, 10}]

(* Andrey Zabolotskiy, Dec 27 2018 *)

CROSSREFS

Sequence in context: A105672 A214271 A262007 * A015152 A021461 A075733

Adjacent sequences:  A005486 A005487 A005488 * A005490 A005491 A005492

KEYWORD

nonn,nice,more

AUTHOR

David J. Thompson

EXTENSIONS

a(11)-a(30) from Andrey Zabolotskiy, Dec 27 2018

STATUS

approved

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Last modified January 21 19:37 EST 2019. Contains 319350 sequences. (Running on oeis4.)